The Milne model was a
special-relativistic cosmological
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
model
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
proposed by
Edward Arthur Milne
Edward Arthur Milne FRS (; 14 February 1896 – 21 September 1950) was a British astrophysicist and mathematician.
Biography
Milne was born in Hull, Yorkshire, England. He attended Hymers College and from there he won an open scholarship ...
in 1935. It is mathematically equivalent to a special case of the
FLRW model in the limit of zero
energy density and it obeys the
cosmological principle
In modern physical cosmology, the cosmological principle is the notion that the spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throu ...
. The Milne model is also similar to
Rindler space, a simple re-
parameterization
In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, d ...
of flat
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
.
Since it features both zero energy density and maximally negative
spatial curvature
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, the Milne model is inconsistent with
cosmological observations. Cosmologists actually observe the universe's
density parameter
The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedma ...
to be consistent with
unity
Unity may refer to:
Buildings
* Unity Building, Oregon, Illinois, US; a historic building
* Unity Building (Chicago), Illinois, US; a skyscraper
* Unity Buildings, Liverpool, UK; two buildings in England
* Unity Chapel, Wyoming, Wisconsin, US; ...
and its curvature to be consistent with
flatness.
Milne metric
The Milne universe is a special case of a more general
Friedmann–Lemaître–Robertson–Walker model (FLRW). The Milne solution can be obtained from the more generic FLRW model by demanding that the energy density, pressure and cosmological constant all equal zero and the spatial curvature is negative. From these assumptions and the Friedmann equations it follows that the scale factor must depend on time coordinate linearly.
[
]
[
]
Setting the spatial curvature and speed of light to unity the metric for a Milne universe can be expressed with hyperspherical coordinates as:
[
]
:
where
:
is the metric for a two-sphere and
:
is the
curvature-corrected radial component for negatively curved space that varies between 0 and
.
The empty space that the Milne model describes can be identified with the inside of a light cone of an event in Minkowski space by a change of coordinates.
Milne developed this model independent of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
but with awareness of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
. As he initially described it, the model has no expansion of space, so all of the redshift (except that caused by
peculiar velocities) is explained by a
recessional velocity
Recessional velocity is the rate at which an extragalactic astronomical object recedes (becomes more distant) from an observer as a result of the expansion of the universe. It can be measured by observing the wavelength shifts of spectral lines e ...
associated with the hypothetical "explosion". However, the mathematical equivalence of the zero
energy density (
) version of the
FLRW metric to Milne's model implies that a full general relativistic treatment using Milne's assumptions would result in an increasing
scale factor
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
and associated
metric expansion of space
The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby the scale of space itself changes. The universe does not ex ...
with the unique feature of a linearly increasing scale factor for all time since the
deceleration parameter
The deceleration parameter ''q'' in cosmology is a dimensionless measure of the cosmic acceleration of the expansion of space in a Friedmann–Lemaître–Robertson–Walker universe. It is defined by:
:q \ \stackrel\ -\frac
where a is the s ...
is uniquely zero for such a model.
Incompatibility with observation
Even though the Milne model as a special case of a Friedmann-Robertson-Walker universe is a solution to General relativity, the assumption of zero energy content limits its use as a realistic description of the universe. Besides lacking the capability of describing matter Milne's universe is also incompatible with certain
cosmological observations. In particular it makes no prediction of the
cosmic microwave background radiation
In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all space ...
nor the
abundance of light elements which are hallmark pieces of evidence that cosmologists agree support Big Bang cosmology over alternatives.
Milne's density function
Milne proposed that the universe's density changes in time because of an initial outward explosion of matter. Milne's model assumes an inhomogeneous density function which is Lorentz Invariant (around the event t=x=y=z=0). When rendered graphically Milne's density distribution shows a three-dimensional spherical
Lobachevskian pattern with outer edges moving outward at the speed of light. Every inertial body perceives itself to be at the center of the explosion of matter (see
observable universe
The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these ob ...
), and sees the local universe as homogeneous and isotropic in the sense of the
cosmological principle
In modern physical cosmology, the cosmological principle is the notion that the spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throu ...
.
Unless the universe modeled has zero density, Milne's proposal does not follow the predictions of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
for the curvature of space caused by global matter distribution, as seen in, for example statistics associated with
large-scale structure.
Differences between Milne model and other models
In order to explain the existence of matter in the universe, Milne proposed a physical
explosion of matter which would not affect the universe's geometry. This is in contrast to the
metric expansion of space
The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby the scale of space itself changes. The universe does not ex ...
that is the hallmark feature of many of the more famous cosmological models including the
Big Bang and
Steady State
In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...
models. Milne's universe shares a superficial similarity to Einstein's
static universe
In cosmology, a static universe (also referred to as stationary, infinite, static infinite or static eternal) is a cosmological model in which the universe is both spatially and temporally infinite, and space is neither expanding nor contract ...
in that the
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
of
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually cons ...
is not
time-dependent. Unlike Einstein's initial cosmology, Milne's proposal directly contradicts the Einstein equations for cosmological scales.
Special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
becomes a global property of Milne's universe while
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
is confined to a local property. The reverse is true for standard cosmological models, and most scientists and mathematicians agree that the latter is self-consistent while the former is mathematically impossible.
Edward Arthur Milne
Edward Arthur Milne FRS (; 14 February 1896 – 21 September 1950) was a British astrophysicist and mathematician.
Biography
Milne was born in Hull, Yorkshire, England. He attended Hymers College and from there he won an open scholarship ...
predicted a kind of
event horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s.
In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact ob ...
through the use of this model: "The particles near the boundary tend towards invisibility as seen by the central observer, and fade into a continuous background of finite intensity." The horizon arises naturally from
length contraction
Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGerald ...
seen in special relativity which is a consequence of the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
upper bound for physical objects. In Milne's universe, the velocities of objects approach this upper bound while the distance to these objects approaches the speed of light multiplied by the time since the event of the initial explosion of material. Beyond this distance, objects do not lie in the observable part of the Milne universe.
At the time Milne proposed his model, observations of the universe did not appear to be in a
homogeneous form. This, to Milne, was a deficiency inherent in the competing cosmological models which relied on the
cosmological principle
In modern physical cosmology, the cosmological principle is the notion that the spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throu ...
that demanded a homogeneous universe. “This conventional homogeneity is only definite when the motion of the particles is first prescribed.” With present observations of the homogeneity of the universe on the largest scales seen in the
cosmic microwave background
In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
and in the so-called "
End of Greatness
The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these ob ...
", questions about the homogeneity of the universe have been settled in the minds of most
observational cosmologists.
Notes
References
Milne Cosmology: Why I Keep Talking About It - a detailed non-technical introduction to the Milne model
* Wegener, Mogens True. Non-Standard Relativity: A Philosopher's Handbook of Heresies in Physics. BoD–Books on Demand, 2016. A thorough historical and theoretical study of the British Tradition in Cosmology, and one long celebration of Milne.
{{Relativity
Obsolete theories in physics
Exact solutions in general relativity
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